On Infinite Series of Bessel functions of the First Kind: ${\sum }_{\nu }{J}_{N\nu +p}(x),{\sum }_{\nu }(-1{)}^{\nu }{J}_{N\nu +p}(x)$

Infinite series of Bessel function of the first kind, ${\sum }_{\nu }^{\pm \mathrm{\infty }}{J}_{N\nu +p}(x)$, ${\sum }_{\nu }^{\pm \mathrm{\infty }}(-1{)}^{\nu }{J}_{N\nu +p}(x)$, are summed in closed form. These expressions are evaluated by engineering a Dirac comb that selects specific sequences within the Bessel series.